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Properties

Label 2-1520-19.11-c1-0-1
Degree $2$
Conductor $1520$
Sign $-0.221 + 0.975i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 + 2.21i)3-s + (−0.5 + 0.866i)5-s − 0.438·7-s + (−1.78 − 3.08i)9-s − 11-s + (1 + 1.73i)13-s + (−1.28 − 2.21i)15-s + (−2.56 + 4.43i)17-s + (2.5 + 3.57i)19-s + (0.561 − 0.972i)21-s + (−2.34 − 4.05i)23-s + (−0.499 − 0.866i)25-s + 1.43·27-s + (−1 − 1.73i)29-s − 10.2·31-s + ⋯
L(s)  = 1  + (−0.739 + 1.28i)3-s + (−0.223 + 0.387i)5-s − 0.165·7-s + (−0.593 − 1.02i)9-s − 0.301·11-s + (0.277 + 0.480i)13-s + (−0.330 − 0.572i)15-s + (−0.621 + 1.07i)17-s + (0.573 + 0.819i)19-s + (0.122 − 0.212i)21-s + (−0.488 − 0.845i)23-s + (−0.0999 − 0.173i)25-s + 0.276·27-s + (−0.185 − 0.321i)29-s − 1.84·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1730277849\)
\(L(\frac12)\) \(\approx\) \(0.1730277849\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-2.5 - 3.57i)T \)
good3 \( 1 + (1.28 - 2.21i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.438T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.56 - 4.43i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.34 + 4.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 4.68T + 37T^{2} \)
41 \( 1 + (-3.06 + 5.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.56 + 2.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.43 - 2.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.78 - 6.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.28 + 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.56 + 4.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.71 - 8.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.12 + 14.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.842 + 1.45i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.56 - 9.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (-1.34 - 2.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.842 - 1.45i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.13093372064668499284482401887, −9.437198728926567339884006820087, −8.607774985083233045395016819805, −7.65477824394315366163176538269, −6.59005265012597755974375691984, −5.84589048318705728931731542334, −5.08304803830223293469458239309, −3.99733703012189420006564544915, −3.63467803888061571266997020742, −2.04998185605813079083741130273, 0.080060210428464532928350605567, 1.21313058010813027384973123323, 2.42450661604398096039379406919, 3.67205528025746210728442493852, 5.07680652642631327101979343945, 5.55905041176864560052761320222, 6.58494977661014021792291703181, 7.28696029580306736226170161730, 7.78992622615567940899210411101, 8.855928863629385804396367926091

Graph of the $Z$-function along the critical line